Q. 65 - Calculus

Question

Every month, Jack hides \(50 under a broken floorboard to save up for a new boat. After t months of saving, he will have $F\left(t\right)=50t$ dollars.

(a) The boat Jack wants costs at least \)7,465. How many months does Jack have to save money before he will have enough to pay for the boat? Illustrate this information on a graph of F(t). 

(b) Suppose a different boat costs M dollars. Will there be a value t equal to N for which $F\left(N\right)>M$? What does this mean in real-world terms? Illustrate the roles of M and N on a graph of F(t).

Step-by-Step Solution

Answer

Part(a) $149.3$ months

Part(b) $N=149.3$

Part(a) Step 1. Given Information

The given function is $F\left(t\right)=50t$.The boat Jack wants costs at least $7,465.

The graph is 

Part (a) Step 2. Explanation

We will first plot the graph with the given data,

Hence, Jack have to save money for $149.3$ months.

Part(b) Step 1. Explanation

Considering the statement, the limit expression can be written as 

$\underset{t\to \infty }{lim}50t=\infty , M=7465$

Here, we have,

$f\left(x\right)=4-x^2, c=\infty , L=-\infty$

Now, for all M positive, there is some N positive such that if $t\in (N,\infty ) \mathrm{then} 50t\in (M,\infty )$

The smallest value of N is given by,

$$\begin{gathered} 50t=7465 \\ t=149.3 \end{gathered}$$

Hence, $N=149.3$